Riemann-Hilbert Problems With Shift On The Lyapunov Curve For Null-solutions Of Iterated Beltrami Equations

This thesis studies Riemann-Hilbert problems with shift on the Lyapunov curve for generali-zed poly-analytic functions called generalized ?-analytic functions of order n,which are nullsolutions of a class of iterated Beltrami equations with order n.Firstly,the Cauchy formula of the generalized ?-analytic function with first order on the unbounded region is established.The compactness of ?-Cauchy integral operators with shift is discussed,Therefore the weakly singular kernel associated with the analytic function with first order is constructed and proved for the first time,and various integral representations of the generalized ?-analytic function are obtained.Then,a new concept of canonical decomposition is introduced and the theory of canonical matrix factorization is developed,especially the theory of matrix factorization for triangular matrix functions in the frame of Beltrami equations.Finally,the Riemann-Hilbert problem with shift is discussed in detail by using the explicit canonical matrix of the triangular matrix function and the decomposition theorem of the higher order ?-analytic function,and then we obtain explicit formulae of solutions and conditions of solvability for this type of problem.The full thesis is divided into six chapters,the content is arranged as follows:The first chapter introduces the research background and research status of boundary value problem for generalized ?-analytic functions,and explains the motives,significance,difficulties in this research work,and some ideas for solving these difficulties.The second chapter mainly introduces some basic knowledge that this thesis needs to use.The third chapter is the first research point of this thesis.Based on the integral representation of the generalized ?-analytic function on the bounded region,this thesis first gives the integral representation of the generalized ?-analytic function on the unbounded region.By constructing shift transformation and estimating techniques of inequalities,the thesis proves that the singular integral operator with shift is a compact operator,and thus gives several weakly singular kernels associated with the generalized ?-analytic function,and based on these kernels the thesis first derive the integral representations of the sectionally generalized ?-analytic functions.The main results not only extend integral representation formulas studied by R Blaya and D Katz,but also laies a solid foundation for the subsequent study of boundary value problems.The fourth chapter is the second research point of this thesis.First,the decomposition theorem of generalized ?-analytic functions with high order by induction is gave.Then,this thesis studies the Riemann-Hilbert problem with shift for generalized ?-analytic function with first order,and obtain the expression and solvable condition of the solution to this problem.Finally,the concept of canonical matrix factorization under the ?-analytic function framework is introduced for the first time,and the corresponding matrix factorization theory is developed,which generalized the classical matrix factorization theory.The fifth chapter is the third research point of this thesis.Using the decomposition theorem and matrix transformation,the Riemann-Hilbert problem with shift of iterative Beltrami equation with the order n is converted into a Riemann-Hilbert problem with shift of the vector-valued Beltrami equation with first order.Then based on the canonical matrix factorization theory of Chapter 4,the general solution and solvable conditions of the original problem are obtained.Our results generalize those of the classical boundary value problems for analytic functions,Riemann and Haseman boundary value problems for polyanalytic functions or generalized ?-analytic functions.The last chapter is the summary and outlook of this thesis.This chapter summarizes the main content of this article and expounds the problems and deficiencies.On this basis,the direction and content that we can continue to study in the future are described.